Recently, there has been an increasing interest in direct applications of parallel mechanisms to real-world industrial problems. In situations where the needs for accuracy and sturdiness dominate the requirement of a large workspace, parallel mechanisms present themselves as viable alternatives to their serial counterparts.
Parallel mechanisms generally comprise two platforms which are connected by a plurality of prismatic joints or legs acting in-parallel. The most common configuration comprises six legs, and the legs are linear actuators such as hydraulic cylinders. One of the platforms is defined as the "movable platform," which has six degrees of freedom relative to the other platform, which is the "base." With six degrees of freedom, the movable platform is capable of moving in three linear directions and three angular directions singularly or in any combination.
When the locations of the connection points of the prismatic legs are known for both platforms, and when a means to change and sense the lengths of the six prismatic legs is provided, then closed-loop control of the position and orientation of the movable platform relative to the base platform is possible and depends on the solutions to two geometrical computations. First, when a desired position and orientation of the movable platform is known, then the corresponding desired lengths of the six legs are determined via a so-called "reverse displacement analysis."
On the other hand, when the actual leg lengths are known, the determination of the actual position and orientation of the movable platform is accomplished via a so-called "forward displacement analysis." Because there are multiple mathematical solutions to this problem (i.e., different possible geometric closures), it is clear that the forward displacement analysis is far more complicated than the reverse analysis. Because of the complexities of such parallel devices, the present inventors believe that for the closed-loop control of the position and orientation of the movable platform, the prior art only successfully utilizes an iterative forward displacement analysis together with the simpler reverse displacement analysis.
An iterative methodology for accomplishing the forward displacement analysis has many drawbacks. First, the solution is not obtained in a prescribed fixed amount of time. Iterative solutions to the mathematics of devices of this type typically are conducted with multidimensional searches, which are extremely time consuming and have no constant search time. For those that study closed-loop control systems, it is well known that timing is critical.
Second, an iterative solution depends on good initial guesses of the position and orientation of the movable platform. To combat this, it is customary to use the last known actual position and orientation as an initial guess to ascertain the "new" actual position and orientation. However when using this technique, it is possible that the mechanism can "run away" from the iterative algorithm. Whenever the legs are able to move faster than the iterative algorithm can manage, the algorithm will have difficulty reporting the actual position and orientation of the movable platform.
Third, an iterative solution has no way of determining regions of mechanism singularities, which are regions where the movable platform cannot be fully constrained no matter what the forces in the legs may be. Singularities such as these cause static instabilities in parallel mechanisms. Multiple closures can exist in these regions. This will cause numerical problems in the iterative algorithm, because the algorithm can actually report the mechanism to be in a wrong closure. Stated in other words, there is no mathematical way of determining closures in an iterative algorithm.
One prior art parallel mechanism is known as a "Stewart" platform. This configuration was introduced in 1965 for use in an aircraft simulator. Since then, numerous parallel mechanisms containing prismatic joints have been called Stewart platforms, although D. Stewart originally suggested only two different arrangements. Stewart's two suggested arrangements were a 3-3 platform (see FIGS. 1 and 2) and a 6-3 platform (see FIGS. 3 and 4). The nomenclature "3-3" signifies three points of connection on the base and three on the movable platform, while "6-3" signifies six points of connection on the base and three on the movable platform.
The Stewart-type parallel mechanisms are defined as those platforms whose six legs meet pair-wise in three points in at least one platform. In other words, the 3-3 and the 6-3 platforms are Stewart platforms. In a so-called 3-3 Stewart platform, there are three connecting points on the base and the three connecting points on the movable platform, with two legs intersecting in each connecting point. Each pair of legs in a 3-3 Stewart platform must be joined by a pair of concentric ball and socket or other type of universal joint which accommodate the endpoints of two legs, a difficult and complicated arrangement which produces unwanted interference between moving parts.
To avoid complications at the joints, it would be desirable to increase the number of connecting points, as is accomplished in the 6-3 platform. The 6-3 platform has six connecting points on the base and three connecting points on the movable platform with join the endpoints of two legs. So, even though the 3-3 platform is geometrically the simplest, it is more desirable from a design point of view to use a 6-3 platform to eliminate three of the six coincident (double) connecting points.
Although drastically increasing the geometrical complexities, designers have extended the Stewart platform by developing 6-6 platforms to completely eliminate the need for coincident connection points. In a 6-6 platform, six connecting points are on both the base and the movable platforms, so that each leg is attached at a distinct point on the base and a distinct point on the movable platform, and no connecting points joint multiple legs. Current industry practice is to choose these six point to be co-planar in each of the platforms such that they form hexagons in each platform. This 6-6 configuration has completely eliminated the need for legs to share connecting points and is now the most widely employed type of device in the flight simulation industry. This industry standard 6-6 platform with hexagonal platforms is referred to here as a "general 6-6," and it is not considered to be a Stewart-type platform because it is much more geometrically complicated.
Much of the research in the literature has devoted extensive effort to the known reverse displacement analysis that is inherently simple for parallel mechanisms (namely, it is required to compute a set of leg lengths given a desired location of the movable platform relative to the base). Thus, the reverse displacement analysis problem statement reads: given a set of vectors expressing the position and orientation of the movable platform, determine the six leg lengths. Thus, this problem is easily solved and applicable to the Stewart-type platforms as well as the industry standard, general 6-6 platform.
Utilizing only a reverse displacement analysis in a given control law constitutes open-loop control of the position and orientation of the movable platform relative to the base. In other words, a controller only commands a desired position and orientation of the movable platform and is not updated periodically as to the actual position and orientation. Hence, the controller's further operation is contingent upon the legs reaching their commanded positions obtained from the reverse displacement analysis.
On the other hand, incorporating a forward displacement analysis "closes the loop" by providing a cartesian controller with feedback information as to the position and orientation of the movable platform. Accordingly, the forward displacement analysis depends on the calculation of the vectors expressing the actual position and orientation of the movable platform, given the actual lengths of the six legs. The forward displacement analysis is far more difficult than the reverse displacement analysis. This stems from the fact that given a set of six leg lengths (and the two platforms), there are a number of different "closures" resulting from the fact that there are a number of different solutions of the vectors representing the position and orientation of the movable platform. There are at least twenty-four real, and probably more, closures to a general 6-6 platform.
On the other hand, there are a maximum of sixteen real closures to the geometrically simpler Stewart-type platforms (i.e., both the 3-3 and 6-3 platforms). The closures are grouped pair-wise due to reflections through the base platform. For example, FIGS. 5A-5F illustrate six different closures for a 3-3 Stewart platform, all having the same leg lengths. It will be readily apparent that only the closure of FIG. 5B, which is a "convex" closure, is a realistic closure in the sense that no legs had to cross each other to arrive at that position and orientation from the configuration of, say, FIG. 1. However, the other five closures, which are "concave" closures, cannot be mathematically distinguished from the convex closure utilizing only the leg lengths. It will also be appreciated that the closures of FIGS. 5A and 5C-5F have crossed legs and are configurationally impossible, yet they represent actual solutions to the mathematical analysis.
Control by a "closed-form forward displacement analysis" greatly differs from an iterative forward displacement analysis because the computation is reduced to a single polynomial in a single variable. The degree of the polynomial is exactly the maximum number of real closures of the mechanism given the lengths of the legs. The control method is considered "closed-form" by kinematicians because the polynomial is obtained directly from known dimensions, even though the polynomial may itself be rooted in an iterative way with a single dimensional search. The Stewart-type platforms (the 3-3 and the 6-3) are characterized by a 16th degree polynomial with even-powered coefficients, while it is not yet know what degree polynomial characterizes the general 6-6; it is at least 24th degree, and most likely more.
A closed-loop controller of a Stewart-type platform or a general 6-6 platform that incorporates a closed-form forward displacement analysis will be greatly enhanced in many ways. First, the implementation will provide a computer-time repeatable and dependable process for determining position and orientation of the movable platform. Second, the implementation can be optimized to run within a parallel computing architecture such that the analysis is accomplished much faster than in the iterative way. Third, the implementation can be used to correctly distinguish closures near singularities.
It is also clear that a closed-form forward displacement analysis will yield much important information on the geometry and kinematics of a parallel mechanism. For instance, a closed-form solution for a Stewart platform will not only yield the exact number of real configurations of the movable platform relative to the base for a specified set of leg lengths, but will also quantify the effects of errors in leg lengths on the position and orientation of the movable platform. Furthermore, control with a closed-form forward displacement controller for a Stewart platform manipulator could provide a Cartesian controller (at a higher control level) with necessary feedback information, namely, the position and orientation of the movable platform relative to the base. This is especially important when the actual position and orientation cannot be directly sensed, and when the manipulator's configuration is determined solely from lengths of the connecting prismatic legs.
Additionally, in the field of force control, a closed-form form forward displacement analysis would provide the necessary analytics to enhance the use of a Stewart platform as a force/torque sensor. A Stewart platform design that is based on an in-parallel structure lends itself well to static force analysis, particularly when utilizing the theory of screws. A wrench applied to the movable platform can be statically equated to the summation of forces measured along the lines of the six prismatic legs. Thus far, this particular application of the Stewart platform has depended on relatively "small" leg deflections, resulting in a "constant" configuration. However, employing a forward displacement analysis provides the analytics to monitor gross deflections of a Stewart platform and thereby permit one to consider the design of a more compliant force/torque sensor. In other words, controller programmed to effect a forward displacement analysis could generate the geometry of the lines of the six connecting prismatic legs of the Stewart platform so that the effects of finite changes in leg lengths can be related to the forces and torques (wrenches) applied to the movable platform.
Accordingly, there is a need in the art to provide a closed-form forward displacement analysis-based controller for the 3-3 and 6-3 Stewart platforms of a simpler geometry, as well as for the general 6-6 platform of a more complex geometry.